Quartile (ungrouped)

MEASURES OF POSITION FOR UNGROUPED DATA:
QUARTILES , DECILES , & PERCENTILES

QUARTILES FOR UNGROUPED DATA
 The QUARTILES are the score points which divide a distribution into four
equal parts.
Q1 Q2 Q3
 25% of the data has a value ≤ Q1

QUARTILES FOR UNGROUPED DATA
Q1 is called the LOWER QUARTILE
Q2 is nothing but the MEDIAN
Q3 is the UPPER QUARTILE

MENDENHALL AND SINCICH METHOD :
A METHOD OF FINDING THE QUARTILE VALUE
 This method is being developed by William Mendenhall and Terry
Sincich to find the position of the quartile in the given data.

 Formula:
Lower Quartile (L) = Position of Q1= ¼ (n+1)
Q2= 2(n+1) = n+1 th observation
4 2
Upper Quartile (U) = Position of Q3 = ¾ (n+1)

 N is the number of elements in the data
Example: The manager of a food chain recorded the number of
customers who came to eat the products in each day. The results
were 10,15,14,13,20,19,12 and 11.
 In this example N=8

 Q1= n+1 th observation
4
 Q2= 2(n+1) = n+1 th observation
4 2
 Q3= 3(n+1) th observation
4

 Steps to solve the quartile of the given data
1. Arrange the data in ascending order or from the lowest value to the
highest value.
2. Find the N or the total number of elements presented in the data.
3. Find the least value of the data and the greatest value of the data.
4. find the lower quartile of the given data using the Mendenhall & Sincich
Method.
Lower Quartile (L) = Position of Q1= ¼ (n+1)

5. Find the middle value of the data or the Median. Use this
formula
Q2= 2(n+1) = n+1 th observation
4 2
6. Find the upper quartile of the given data. Use this formula
Upper Quartile (U) = Position of Q3 = ¾ (n+1)

 Example:
The owner of the coffee shop recorded the number of customers
who came into his café each hour in a day. The results were 14, 10,
12, 9, 17, 5, 8, 9, 14, 10, and 11.

 Solution
 Ascending order
{5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 N=11
 Least value= 5
 Greatest value= 17

 Lower Quartile (L) = Position of Q1= ¼ (n+1)
Q1= ½ (n+1)
Q1= ½ (11+1)
Q1= ½ (12)
Q1= 12/4 (divide)
Q1= 3
{5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 Therefore the Q1 is the 3rd element in the data which is 9

 Median Value or the middle value
Q2= 2/4 (n+1) = n+1/2 th observation
Q2= 2/4 (11+1)
Q2= 2/4 (12)
Q2= 24/4
Q2= 6 {5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 Therefore the Q2 is the 6th element in the data which is 10

 Upper Quartile (U)= Position of Q3= ¾ (n+1)
Q3= ¾ (11+1)
Q3= ¾ (12)
Q3= 36/4
Q3= 9 {5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17}
 Therefore the Q3 is the 9th element in the data which is 14

LINEAR INTERPOLATION
 A method of finding the quartile value.
 Is a method of constructing new data points within the range of a
discrete set of known data points.
 It is often required to interpolate, i.e., estimate the value of that
function for an intermediate value of the independent variable.
 We need to use the interpolation if the value of the position is in
decimal form.

 Using the formula
Qk= k/4 (n+1)
k= position of the quartilen= total no. of the data

 Steps in interpolation method
1. Arrange the scores in ascending order
2. Locate the position of the score in the distribution
3. Since the result is in decimal number, proceed to linear
interpolation
4. Find the difference between the two values wherein Q1 is
situated
5. Multiply the result in step 2 by the decimal part obtained in
step 4
6. Add the result in step 5 to the second smaller number in step 4

 Example:
Find the first quartile (Q1), and the third quartile (Q3), Given the
scores of 9 students in their mathematics activity using linear
interpolation
{1, 27, 16, 7, 31, 7, 30, 3, 21 }

 Step 1 : Arrange the scores in ascending order
{ 1, 3, 7, 7, 16, 21, 27, 30, 31 }
 Step 2: Locate the position of the score in the distribution
Position of Q1= ¼(n+1)
Q1= ¼ (9+1)
Q1= 0.25(10)
Q1= 2.5

 Step 3: Since the result is in decimal number, proceed to linear
interpolation
 Step 4: Find the difference between the two values wherein Q1 is
situated
{ 1, 3, 7, 7, 16, 21, 27, 30, 31 }
2.5 position
 Q1 is between the values 3 and 7, therefore
= 7-3
= 4

 Step 5: Multiply the result in step 2 by the decimal part obtained in
step 4
= 4 (0.5)
= 2
 Step 6: Add the result in step 5 to the second smaller number in
step 4
= 2+3
= 5
 Therefore the value of Q1 is equal to 5

Quartile (ungrouped)

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